Hydrodynamic Stability and Transition to Turbulanc

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Description:

Objective:

Fluid flow phenomenon in most of its manifestations, either in daily life or in scientific and engineering studies, is dominated by turbulence. In many of these cases, the phenomenon of turbulence is preceded by a sequence of instabilities, which begin from a simple flow behaviour to progressively more and more complex one. These initial developments could be very well analysed within the framework of hydrodynamic instability, and so, great insights into the cause of such phenomenon and their understanding could be achieved. In certain problems, the subsequent appearance of complexity could be related to the development of chaotic behaviour in typical dynamical systems. The present course aims at the discussion of some of the very common and useful methods in the area of hydrodynamic stability. In addition to shear flows, which are studied a bit more elaborately, the course includes an introduction to thermal and centrifugal instabilities. Brief discussion of nonlinear stability and transition to turbulence from the dynamical systems point of view also appears towards the end of the course. As such, it is hoped that the course will be of immense importance and value to students pursuing almost any field in fluid mechanics and heat transfer. The related discussion of nonlinear dynamics and route to chaos could be of help to students even from other areas in science and engineering.

Course Content

Introduction: overview of methods of stability analysis (linear and nonlinear theories), normal mode analysis; Common examples of gravitational, thermal and centrifugal instabilities: Rayleigh-Taylor instability, Rayleigh-Bénard instability, instabilities in Taylor–Couette flow; Instabilities in parallel flows: temporal instability, Rayleigh equation, Orr-Sommerfeld equation and Squire’s transformation, numerical technique for solving the Orr-Sommerfeld equation, spatial instability and Gaster’s transformation; Spatio-temporal instability: evolution of perturbations in space-time, absolute and convective instabilities, Brigg's method and the pinch point criteria; Stability of non-parallel flows: instabilities in weakly nonparallel flows, parabolised stability equations, bi-global and tri-global stability analysis; Nonlinear stability theory: weak nonlinearity, Landau equation, interaction of linear modes, energy methods; Transition to turbulence: transient growth, pseudo spectra and nonmodal instabilities, receptivity, secondary instability, Morkovin map of roads to wall turbulence, regimes of laminar-turbulent transition in boundary layer flows, routes to chaos and turbulence.

Textbooks:

[1] P. G. Drazin, Introduction to Hydrodynamic Stability, Cambridge University Press, 2002.

[2] P. J. Schmid, D. S. Henningson, Stability and Transition in Shear Flows, Springer-Verlag, 2001.

[3] P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Second Edition, Cambridge University Press, 2004.

References:

[1] Francois Charru, Hydrodynamic Instabilities, Cambridge University Press, 2011.

[2] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, 1961.

[3] C. Godrèche, P. Manneville, Hydrodynamics and Nonlinear Instabilities, Cambridge University Press, 1998.

[4] A. Georgescu, Hydrodynamic stability theory, Springer, 1985.

[5] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press, 2014.

[6] Akiva M. Yaglom, Hydrodynamic Instability and Transition to Turbulence, Springer, 2012.